The 3-rd unramified cohomology for norm one torus

Abstract

For an algebraic torus $S$, Blinstein and Merkurjev have given an estimate of $3$-th unramified cohomology $\bar{H}^3_{nr}(F(S),\mathbb Q/\mathbb Z(2))$ obtained from a flasque resolution of $S$. Based on their work, for the norm one torus $W=R_{K/F}^{(1)}\mathbb{G}_m$ with $K/F$ abelian, we compute the $3$-th unramified cohomology $\bar{H}^3_{nr}(F(W),\mathbb Q/\mathbb Z(2))$.